Projective phase measurements in one-dimensional Bose gases

TL;DR

This paper links time-of-flight measurements in split one-dimensional Bose gases to the eigenvalues of vertex operators, enabling access to multi-point correlation functions and properties of the symmetric sector in experiments.

Contribution

It provides a theoretical framework connecting single-shot interference patterns to multi-point correlations of vertex operators in 1D Bose gases.

Findings

Single measurements relate to eigenvalues of vertex operators.

Derived an expression for particle density in terms of convolutions of vertex operator eigenvalues.

Accessible properties of the symmetric sector through experimental data analysis.

Abstract

We consider time-of-flight measurements in split one-dimensional Bose gases. It is well known that the low-energy sector of such systems can be described in terms of two compact phase fields $\hat{\phi}_{a,s}(x)$. Building on existing results in the literature we discuss how a single projective measurement of the particle density after trap release is in a certain limit related to the eigenvalues of the vertex operator $e^{i\hat{\phi}_a(x)}$. We emphasize the theoretical assumptions underlying the analysis of "single-shot" interference patterns and show that such measurements give direct access to multi-point correlation functions of $e^{i\hat{\phi}_a(x)}$ in a substantial parameter regime. For experimentally relevant situations, we derive an expression for the measured particle density after trap release in terms of convolutions of the eigenvalues of vertex operators involving both sectors of the two-component Luttinger liquid that describes the low-energy regime of the split condensate. This opens the door to accessing properties of the symmetric sector via an appropriate analysis of existing experimental data.